Find the image of the line x+y=1 under the transformation W=z^2

To determine the image of the line \( x + y = 1 \) under the transformation \( W = z^2 \), let's break it down into steps.

### 1. Setup the Transformation

The transformation is given as:

\[
W = z^2 \implies W = u + iv \text{ and } z = x + iy
\]

This implies:

\[
W = (x + iy)^2 = x^2 - y^2 + 2xyi
\]

So, the real part \( u \) and the imaginary part \( v \) of \( W \) are:

\[
u = x^2 - y^2 \quad \text{and} \quad v = 2xy
\]

### 2. Describe the Line \( x + y = 1 \)

The line equation \( x + y = 1 \) can be rewritten as:

\[
y = 1 - x
\]

Substituting \( y = 1 - x \) into the expressions for \( u \) and \( v \):

\[
u = x^2 - (1 - x)^2
\]

\[
v = 2x(1 - x)
\]

### 3. Simplify the Equations

First, simplify \( u \):

\[
u = x^2 - (1 - 2x + x^2) = x^2 - 1 + 2x - x^2 = 2x - 1
\]

Next, simplify \( v \):

\[
v = 2x(1 - x) = 2x - 2x^2
\]

### 4. Parametric Equations for the Image

The parametric equations describing the image are:

\[
u = 2x - 1
\]

\[
v = 2x - 2x^2
\]

Here, \( x \) is the parameter. As \( x \) varies, the corresponding points \( (u, v) \) describe the curve in the \( uv \)-plane.

### 5. Analyze the Curve

The equation \( v = 2x - 2x^2 \) represents a parabola that opens downward. The equation \( u = 2x - 1 \) is linear, mapping \( x \) directly to \( u \).

### 6. Conclusion: Shape of the Image

The image of the line \( x + y = 1 \) under the transformation \( W = z^2 \) is a parabolic curve in the \( uv \)-plane. Specifically, it is given by the parametric curve:

\[
v = \frac{(u + 1)^2}{2} - 1
\]

Would you like more details or further exploration?

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### Related Questions:

1. What is the image of the line \( y = mx + c \) under \( W = z^2 \)?
2. How does the transformation \( W = z^2 \) affect circles?
3. What is the inverse transformation for \( W = z^2 \)?
4. Can the transformation \( W = z^2 \) be used to map entire regions?
5. What happens to horizontal and vertical lines under the transformation?

### Tip:
Understanding how complex transformations like \( W = z^2 \) work can help you visualize conformal mappings and their applications in physics and engineering.

Explore the transformation of the line x + y = 1 under the complex transformation W = z^2. Understand how this transformation maps the line onto a parabolic curve in the uv-plane. Detailed step-by-step analysis with parametric equations provided.

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